Stochastic stabilization and destabilization of hybrid SDEs driven by Lévy noise

Abstract This research aims to investigate the stochastic stabilization and destabilization of hybrid stochastic differential equations (SDEs) with Lévy noise. First, the sufficient conditions for stability and instability of hybrid SDEs systems with Lévy noise are given. Based on these, this paper is then devoted to the effects of the noise (Brownian motion noise and Lévy noise) on stability and instability of hybrid differential equation system. For the more comprehensive analysis, we establish the criteria for the stable and unstable behaviours of the general linear hybrid SDEs systems perturbed by Lévy noise. Last, we reveal a fact that only Lévy noise or Brownian motion noise and Lévy noise together cannot destabilize the stability of a 1D linear stable system, the similar effect occurred in Mao et al. (2007), which states that only Brownian motion noise cannot destroy the stability of a 1D linear stable system.